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Sequences of pyramidal numbers (1).

Abstract Shyam Sunder Gupta [4] has defined Smarandache consecutive and reversed Smarandache sequences of Triangular numbers. Delfim F.M.Torres and Viorica Teca [1] have further investigated these sequences and defined mirror and symmetric Smarandache sequences of Triangular numbers making use of Maple system. One of the authors A.S.Muktibodh [2] working on the same lines has defined and investigated consecutive, reversed, mirror and symmetric Smarandache sequences of pentagonal numbers of dimension 2 using the Maple system. In this paper we have defined and investigated the s-consecutive, s-reversed, s-mirror and s-symmetric sequences of Pyramidal numbers (Triangular numbers of dimension 3.) using Maple 6.

[section] 1. Introduction

Figurate number is a number which can be represented by a regular geometrical arrangement of equally spaced points. If the arrangement forms a regular polygon the number is called a polygonal number. Different figurate sequences are formed depending upon the dimension we consider. Each dimension gives rise to a system of figurate sequences which are infinite in number.

In this paper we consider a figurate sequence of Triangular numbers of dimension 3, also called as Pyramidal numbers.

The nth Pyramidal number [t.sub.n], n [member of] N is defined by:

[t.sub.n] = n(n+1)(n+2) / 6

We can obtain the first k terms of Pyramidal numbers in Maple as;
> t := n->(1/6)*n*(n+1)*(n+2):
> first :+ k -> seq (t(n), n=1 ... 20):
> first(20);

 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680,
 816, 969, 1140, 1330, 1540


Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)

Combining(17), (19) and (20), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which has a Dirichlet series expansion, absolutely convergent for [??]s > 1/5.

By Lemma 1, for Rs > 1 we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

For constructing Smarandache sequence of Pyramidal numbers we use the operation of concatenation on the terms of the above sequence.This operation is defined as ;

> conc :=(n,m)-> n*10^length(m)+m:

We define Smarandache consecutive sequence {[scs.sub.n]} for Pyramidal numbers recursively as;
[scs.sub.1] = [u.sub.1],
[scs.sub.n] = conc([scs.sub.n-1], [u.sub.n])


Using MapleWe have obtained first 20 terms of Smarandache consecutive sequence of Pyramidal numbers;
>conc :=(n,m)-> n*10^length(m)+m:
> scs_n := (u,n)-> if n = 1 then u(1)else conc(scs_n(u,n-1),u(n))fi:
> scs := (u,n)-> seq (scs_n(u,i),i=1 ... n):
> scs(t,20);

 1, 14, 1410, 141020, 1402035, 1410203556, 141020355684,
 141020355684120, 141020355684120165, 141020355684120165220,
 141020355684120165220286, 141020355684120165220286364,
 141020355684120165220286364455,
 14102035568412016522028636445560,
 14102035568412016522028636445560680,
 14102035568412016522028636445560680816,
 14102035568412016522028636445560680816969,
 141020355684120165220286364455606808169691140,
 1410203556841201652202863644556068081696911401330,
 14102035568412016522028636445560680816969114013301540,


Display of the same sequence in the triangular form is;
> show := L -> map(i ->print(i),L):
> show([scs(t,20)]);

 1
 14
 1410
 141020
 14102035
 1410203556
 141020355684
 141020355684120
 141020355684120165
 141020355684120165220
 141020355684120165220286
 141020355684120165220286364
 141020355684120165220286364455
 141020355684120165220286364455560
 141020355684120165220286364455560680
 141020355684120165220286364455560680816
 141020355684120165220286364455560680816969
 1410203556841201652202863644555606808169691140
 14102035568412016522028636445556068081696911401330
141020355684120165220286364455560680816969114013301540


The reversed Smarandache sequence (rss)associated with a given sequence {[u.sub.n]}; n [member of] N is defined recursively as:
[rss.sub.1] = [u.sub.1],
[rss.sub.n] = conc([u.sub.n], [rss.sub.n-1]).


In Maple we use the following program;
> rss_n :=(u,n) -> if n=1 then u(1) else conc(u(n),rss_n(u,n-1)) fi:
> rss := (u,n) -> seq(rss_n(u,i),i=1..n):


We get the first 20 terms of reversed smarandache sequence of Pyramidal numbers as;
> rss(t,20);

 1, 41, 1041, 201041, 35201041, 5635201041, 845635201041,
 120845635201041, 165120845635201041, 220165120845635201041,
 286220165120845635201041, 364286220165120845635201041,
 455364286220165120845635201041,
 560455364286220165120845635201041,
 680560455364286220165120845635201041,
 816680560455364286220165120845635201041,
 969816680560455364286220165120845635201041,
 1140969816680560455364286220165120845635201041,
 13301140969816680560455364286220165120845635201041,
 154013301140969816680560455364286220165120845635201041


Smarandache Mirror Sequence (sms)is defined as follows:
[sms.sub.1] = [u.sub.1],
[sms.sub.n] = conc(conc([u.sub.n], [sms.sub.n-1]), [u.sub.n]).


The following program gives first 20 terms of Smarandache Mirror sequence of Pyramidal numbers.
> sms_n := (u,n) -> if n=1 then
> u(1)
> else
> conc(conc(u(n),sms_n(u,n-1)),u(n))
> fi:
> sms :=(u,n) ->seq(sms_n(u,i), i=1..n):
> sms(t,20);

 1, 414, 1041410, 20104141020, 352010414102035, 5635201041410203556,
 84563520104141020355684, 12084563520104141020355684120,
 16512084563520104141020355684120165,
 22016512084563520104141020355684120165220,
 28622016512084563520104141020355684120165220286,
 36428622016512084563520104141020355684120165220286364,
 45536428622016512084563520104141020355684120165220286364455,
 5604553642862201651208456352010414102035568412016522028636 4455560, 68056045536428622016512084563520104141020355684120 165220286364455560680, 816680560455364286220165120845635201 04141020355684120165220286364455560680816. 9698166805604553 6428622016512084563520104141020355684120165220286364455560 680816969, 114096981668056045536428622016512084563520104141 0203556841201652202863644555606808169691140, 13301140969816 6805604553642862201651208456352010414102035568412016522028 636445556068081696911401330, 154013301140969816680560455364 2862201651208456352010414102035568412016522028636445556068 0816969114013301540


Finally Smarandache Symmetric sequence (sss) is defined as:
[sss.sub.2n-1] = conc(bld([scs.sub.2n-1]), [rss.sub.2n-1]),
[sss.sub.2n] = conc([scs.sub.2n], [rss.sub.2n]), n [member of] N,


where the function "bld" (But Last Digit) is defined in Maple as

> bld := n->iquo(n,10):

First 20 terms of Smarandache Symmetric sequence are obtained as
> bld :=n-> iquo(n,10):
> conc := (n,m)-> n*10^length(m)+m:
> sss_n := (u,n) -> if type(n,odd) then
> conc(bld(scs_n(u,(n+1)/2)),rss_n(u,(n+1)/2))
> else
> conc(scs_n(u,n/2),rss_n(u,n/2)
> fi:
> sss := (u,n) -> seq(sss_n(u,i), i=1..n):
> sss(t,20);

 1, 11, 141, 1441, 1411041, 14101041, 14102201041, 141020201041,
 141020335201041, 1410203535201041, 1410203555635201041,
 14102035565635201041, 14102035568845635201041,
 141020355684845635201041, 14102035568412120845635201041,
 141020355684120120845635201041,
 14102035568412016165120845635201041,
 141020355684120165165120845635201041,
 14102035568412016522220165120845635201041,
 141020355684120165220220165120845635201041


We find out primes from a large (first 500) terms of various Smarandache sequences defined so far. We have used Maple 6 on Pentium 3 with 128Mb RAM. We first collect the lists of first 500 terms of the consecutive, reversed, mirror and symmetric sequences of Pyramidal numbers;
> st :=time(): Lscs500:=[scs(t,500)]: printf("%a seconds",
round(time()-st)); 15 seconds
> st :=time(): Lrss500:=[rss(t,500)]: printf("%a seconds",
round(time()-st)); 20 seconds
> st :=time(): Lsms500:=[sms(t,500)]: printf("%a seconds",
round(time()-st)); 58 seconds
> st :=time(): Lsss500:=[sss(t,500)]: printf("%a seconds",
round(time()-st)); 12 seconds


Further we find the number of digits in the 500th term of each sequence.
> length(Lscs500[500]),length(Lrss500[500]);

3283, 3283

> length(Lsms500[500]),length(Lsss500[500]);

6565, 2846


There exist no prime in the first 500 terms of Smarandache consecutive sequence of Pyramidal numbers;
> st:= time():select(isprime,Lscs500);

[]

> printf("%a minutes",round((time()-st)/60));
9 minutes


There is only one prime in the first 500 terms of reversed Smarandache sequence of Pyramidal numbers;
> st:= time():
> select(isprime,Lrss500);

[41]

> printf("%a minutes",round((time()-st)/60));
119 minutes


There is no prime in the first 500 terms of Smarandache mirror sequence;
> st:= time():
> select(isprime,Lsms500);
> printf("%a minutes",round((time()-st)/60));

[]

177 minutes


There is only one prime in the first 500 terms of Smarandache symmetric sequence;
> st:= time():
> select(isprime,Lsss500);

[11]

> printf("%a minutes",round((time()-st)/60));
90 minutes


[section] 2. Open problems

1) How many Pyramidal numbers are there in the first 500 terms of Smarandache consecutive, mirror, symmetric and reverse symmetric sequences of Pyramidal numbers ?

2) What are those numbers ?

Received Nov. 12, 2006

References

[1] Delfim F.M., Viorica Teca, Consective, Reversed, Mirror and Symmetric Smarandache Sequences of Triangular Numbers, Scientia Magna, 1(2005), No.2, 39-45.

[2] Muktibodh A.S., Smarandache Sequences of Pentagonal Numbers, Scientia Magna, 2(2006), No.3.

[3] Muktibodh A.S., Figurate Systems, Bull. Marathwada Mathematical Soc., 6(2005), No.1, 12-20.

[4] Shyam Sunder Gupta, Smarandache Sequence of Triangular Numbers, Smarandache Notions Journal, 14, 366-368.

Arun Muktibodh, Uzma Sheikh, Dolly Juneja, Rahul Barhate and Yogesh Miglani

Mohota Science College

Umred Rd., Nagpur-440009, India.
COPYRIGHT 2007 American Research Press
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Author:Muktibodh, Arun; Sheikh, Uzma; Juneja, Dolly; Barhate, Rahul; Miglani, Yogesh
Publication:Scientia Magna
Geographic Code:1USA
Date:Jan 1, 2007
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